MindMap Gallery 031 Number Operations Integer Operations Mind Map
Primary school mathematics review mind map on operations of integers, including the four arithmetic operations of integers, division of integers, addition of integers, subtraction of integers, multiplication of integers, etc.
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Number operations: Integer operations
Addition of integers
The meaning of addition
The operation of combining two numbers into one number is called addition.
Addends and sums: The two numbers added are called addends, and the resulting number is called the sum.
Operation rules for vertical addition
When adding multiple digits to multiple digits, the same digits should be aligned, and starting from the ones digit, the number on which digit adds up to ten is like moving forward one digit.
The law of addition
commutative law of addition
When two numbers are added, the positions of the addends are swapped, and the sum remains unchanged. a b=b a
additive associative law
To add three numbers, add the first two numbers first, or add the last two numbers first, and the sum remains unchanged. (a b) c=a (b c)
application
Appropriate use of the commutative law of addition and the associative law can reduce the difficulty of calculation.
The changing pattern of sum
and changes
If one addend increases (or decreases) by a number and the other addend remains unchanged, then their sum also increases (or decreases) by the same amount. a b=c ,a m b=c m, a-m b=c-m , (a m) (b-n)=c m-n,
and unchanged
If one addend increases (or decreases) by a number and another addend decreases (or increases) by the same amount, then their sum does not change. (a m) (b-m)=c
Subtraction of integers
The meaning of subtraction
Given the sum of two addends and one of the addends, find the other addend
Rules for subtracting vertical expressions
When subtracting multiple digits from a multi-digit number, you should align the same digits of the minuend and the subtrahend, and then start subtracting from the ones digit. If the number in any digit of the minuend is not enough to subtract, you should step back one from the previous digit and make it ten. , added together with the number on the base and then subtracted, this is called abdication, also called misalignment.
Subtraction checking: You can use addition checking and subtraction checking.
Operational properties of subtraction
Continuous reduction
Subtracting two consecutive numbers from a number is equal to this number minus the sum of the two numbers. a-b-c=a-(b c)
Subtract the difference between two numbers
Subtracting the difference of two numbers from a number is equal to subtracting the minuend from the difference and then adding the subtrahend to the number. a-(b-c)=a-b c
Requirements for removing brackets
When removing parentheses, if there is a minus sign outside the parentheses, the sign inside the parentheses will change; if there is a plus sign outside the parentheses, the sign inside the parentheses will not change.
Bad change pattern
difference change
If the minuend increases (or decreases) by a number and the minuend does not change, then their difference also increases (or decreases) by the same amount. If a-b=c then (a±m)-b=c ±m
If the minuend and minuend remain unchanged and the subtrahend increases (or decreases) by a number, then their difference decreases (or increases) by the same amount.
Difference remains unchanged
If the minuand and subtrahend increase (or decrease) by the same amount at the same time, their difference remains unchanged.
The relationship between addition and subtraction
relation
Subtraction is the inverse operation of addition
The relationship between the parts of addition and subtraction
Addend Addend = sum, and - addend = another addend.
Minuend - Minuend = Difference, Minuend - Difference = Minuend, Difference Minuend = Minuend
Using the above relations, you can perform calculations of addition and subtraction.
Multiplication of integers
The meaning of integer multiplication
significance
The simple operation of finding the sum of several identical addends is called multiplication.
expression
Factor × factor = product a × b =c
Vertical operation rules for integer multiplication
law
Align the end, multiply from the ones digit, multiply the number in each digit of the factor by another factor; multiply by the number in which digit of the factor, the last digit of the resulting product will be aligned with that digit, and finally multiply the number of times The products of the multiplications add up.
Check calculation
Product ÷ one of the factors = the other factor; use the commutative law of multiplication to exchange the positions of the two factors and calculate again.
Laws of operation of multiplication
commutative law of multiplication
When two numbers are multiplied together, the positions of the two numbers are swapped, and the product remains unchanged. a×b=b×a
Note: The commutative law of multiplication applies to expressions that only contain multiplication operations.
multiplicative associative law
To multiply three numbers, multiply the first two numbers first, or multiply the last two numbers first, and the product remains unchanged. (a×b)×c=a×(b×c)
distributive law of multiplication
law
If the sum of two numbers is multiplied by a number, you can first multiply them by the number and then add them together, and the result will not change. (a b)×c=a×c b×c. This law can be applied not only in a positive direction, but also in a reverse direction.
application
Calculations can be simplified using the rules of operations. The principle of using the associative and commutative laws is to multiply the factors to get the numbers of whole tens, whole hundreds, and whole thousand.
The changing pattern of product
product change
One factor remains unchanged, the other factor multiplies (or divides) a number that is not 0, and their product also multiplies (or divides) this number. a ×b=c, a÷m×b=c÷m , a×m×b=c×m (m≠0)
The product remains unchanged
When one factor multiplies (or divides) a number that is not 0, and another factor divides (or multiplies) that number, their product remains unchanged. a ×b=c, a×m×(b÷m)=c (m≠0)
Division of integers
The meaning of integer division
significance
If the product of two factors and one of the factors is known, the operation of finding the other factor is called division. Division is the inverse operation of multiplication.
Precautions
When 0 is divided by any number that is not 0, the quotient is 0, that is, 0÷a=0 a≠0
It is necessary to distinguish between dividing and dividing by. For example: 9÷3 can be read as 9 divided by 3, or as 3 divided by 9.
Division with remainder
Division with remainder
Dividend ÷ divisor = quotient.....remainder. For example: 14÷5=2.....4
In integer division with a remainder, the remainder must be smaller than the divisor. The largest remainder is 1 less than the divisor.
relationship between parts
Divisor = Quotient × Divisor Remainder; Divisor = (Divisor - Remainder) ÷ Quotient
"Divisor = quotient × divisor remainder" is often used to check division formulas with remainders.
Arithmetic rules and properties of integer division
Arithmetic rules for vertical division
Starting from the highest digit of the dividend, the number of the divisor depends on the first few digits of the dividend. If the first few digits are smaller than the divisor, then take one more digit and divide. Which digit is divided to, the quotient is written there. One of the top.
If the quotient of any digit of the dividend is less than 1, then write "0" in that digit.
The remainder of each division must be smaller than the divisor.
Trial business methods
When the divisor is close to a whole ten or a hundred, the divisor is usually treated as a whole ten or a hundred to test the quotient.
Test the quotient by treating the dividend and divisor together as positive tens, whole hundreds, or tens and hundreds.
Operation properties of division
A number divided by two numbers is equal to the number divided by the product of the two numbers. a÷b÷c=a÷(b×c) b≠0,c≠0.
The changing pattern of quotient
The quotient remains unchanged
In division, the dividend and divisor are simultaneously (or divided by) the same number (except 0), and the quotient remains unchanged. (a×c)÷(b×c)=a÷b (c≠0); (a÷c)÷(b÷)=a÷b (c≠0)
In division with a remainder, if the dividend and the divisor are at the same time (or divided by) the same number (except 0), the quotient remains unchanged, but their remainders must be multiplied (or divided) by this number.
Quotient changes
If the dividend is multiplied (or divided) by a number other than 0 and the divisor remains unchanged, then their quotient must also be multiplied (or divided) by the same number. a÷b=c ,( a×n)÷b=c×n ; (a÷n)÷b=c÷n (n≠0)
If the dividend remains unchanged and the divisor is multiplied (or divided) by a number other than 0, and the divisor remains unchanged, then their quotient will be divided (or multiplied) by the same number. a÷b=c, a÷(b×n)=c÷n; a÷(b÷n)=c×n (n≠0)
The relationship between multiplication and division
relation
Division is the inverse operation of multiplication.
expression
Factor × factor = product, factor = product ÷ factor, dividend ÷ divisor = quotient, dividend ÷ quotient = divisor, dividend = quotient × divisor.
Multiplication and division calculations
multiplication check
Check using the commutative law of multiplication
Check with division
division calculation
Check with multiplication
Check with division: Swap the divisor and quotient.
Calculation of division with remainder: dividend = divisor × quotient remainder.
Four arithmetic operations on integers
definition
Arithmetic
Addition, subtraction, multiplication, and division are collectively called the four arithmetic operations.
Four mixed operations
In a calculation, two or more of the four operations of addition, subtraction, multiplication, and division are called four mixed operations.
First and second level operations
Among the four mixed operations, addition and subtraction are called first-level operations, and multiplication and division are called second-level operations.
order of operations
Calculation without parentheses
Only addition and subtraction or only multiplication and division: Calculate from left to right.
There are both multiplication and division, as well as addition and subtraction: multiplication and division first, then subtraction.
The formula has parentheses
Only parentheses: first calculate what is inside the parentheses, then what is outside the parentheses.
There are parentheses and square brackets: first calculate what's inside the small brackets, then what's inside the square brackets, and finally what's outside the square brackets.
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